3.1.5 Impact Ionization

Impact ionization is a typical non-equilibrium process which requires a large electric field. An electron (or hole) in the conduction (or valence) band gains its energy by external electric fields and becomes so highly energetic that it can create an electron-hole pair by colliding with an electron in the valence band and exciting it to the conduction band. Under electric fields the total generation rate of electron-hole pairs due to impact ionization is given by

$$G_{ii} = \alpha_n\, \frac{\vert\mathbf{J}_n\vert}{q} + \alpha_p\, \frac{\vert\mathbf{J}_p\vert}{q}.$$ (3.22)

Generation due to impact ionization is proportional to the current density. The electron and hole impact ionization rates $\alpha_n$ and $\alpha_p$ are defined as the number of electron-hole pairs generated by the carriers traveling unit distance along the direction of the electric field. Based on extensive theoretical and experimental investigations of the ionization coefficients in silicon, their variation with electric field has been found to be given by the following empirical relationships

$$\alpha_n = A_n \mathrm{exp}\,\biggl(-{ \biggl(\frac{B_n}{E}\biggr)}^{\beta_n}\biggr)\,,$$ (3.23)

$$\alpha_p = A_p \mathrm{exp}\,\biggl(-{\biggl(\frac{B_p}{E}\biggr)}^{\beta_p}\biggr)\,.$$ (3.24)

There is much debate over the appropriate choices of ionization rate parameters $A_n$, $A_p$, $B_n$, $B_p$, $\beta_n$, and $\beta_p$. For our device simulations, we used the values of van Overstraeten and de Man [117].

Figure 3.4: Impact ionization coefficients in silicon.

As shown in Figure 3.4, it is important to note that the ionization coefficients increase very rapidly with increasing electric field. Due to this strong dependence of the ionization coefficients on the electric field, the breakdown voltage of devices can be severely reduced by the presence of a high localized electric field within the device. For some cases it is enough to take only $\alpha_\mathrm{n}$ into account when solving Poisson's equation for the calculation of reverse characteristics, because $\alpha_\mathrm{n}$ in the silicon is one order higher than $\alpha_\mathrm{p}$. The expressions of the impact ionization coefficients can be approximated in many cases by one ionization coefficient that is useful for deriving analytical solutions for the breakdown voltage calculation. Applying Fulop's power series [118] an approximation for the impact ionization coefficient is

$$\alpha_\mathrm{eff} = a_\alpha\, E^7.$$ (3.25)

$\alpha _\mathrm{eff}$ is known as effective impact ionization coefficient, and $a_\alpha$ is approximately 1.8 $\times$ $10^{-35}$ $\mathrm{cm}^{6}$ $\mathrm{V}^{-7}$ for the silicon. This expression is useful to derive a closed-form solution for the breakdown voltage of abrupt and linearly graded junctions.